Question: Since $2010$, the town of Fall River has been experiencing a growth in population. The relationship between the elapsed time, $t$, in years, since $2010$ and the town's population, $P(t)$, is modeled by the following function. P ( t ) = 36,800 ⋅ 2 t 25 P(t)=36{,}800\cdot 2\^{{\frac{t}{25}}} According to the model, what will the population of Fall River be in $2020$ ? Round your answer, if necessary, to the nearest whole number.
Answer: Thinking about the problem We want to find the population of Fall River in $2020$, which is $10$ years since $2010$. In other words, we are given a $t$ value of $10$ years and want to find the population associated with that input, or $P(10)$. To do this, we can substitute ${10}$ in for $ t$ and evaluate. P ( 10 ) = 36,800 ⋅ 2 10 25 P({10})=36{,}800\cdot 2\^{{\frac{{10}}{25}}} Evaluating the expression We can use a calculator to evaluate the expression. The answer is shown below. P ( 10 ) = 36,800 ⋅ 2 10 25 = 36,800 ⋅ 2 0.4 ≈ 48,558 \begin{aligned}P(10)&=36{,}800\cdot 2\^{{\frac{{10}}{25}}}\\\\ &=36{,}800\cdot 2^{{0.4}}\\\\ &\approx48{,}558\\\\ \end{aligned} In $2020$, the population of Fall River will be about $48{,}558$ people.